English

Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps I

Spectral Theory 2026-03-27 v4 Geometric Topology

Abstract

In this series of articles, we analyse the level-sets of length functions on the moduli space of compact hyperbolic surfaces of fixed genus. This work ultimately culminates in a proof that typical hyperbolic surfaces have an optimal spectral gap. In this first article, we introduce new volume functions VgT(l)V_g^T(l), counting the expected number of closed geodesics of type TT and length ll on a random hyperbolic surface of genus gg. So far, this function has only been considered in the case where the type is simple, in which case it can be expressed as a combination of Weil-Petersson volumes polynomials, as proven by Mirzakhani. We provide an integral expression for VgT(l)V_g^T(l) for any prescribed type TT, which we use to prove that VgT(l)V_g^T(l) admits a full asymptotic expansion in powers of 1/g1/g. We then claim that the coefficients in this expansion, as a function of the length variable ll, belong to a newly-introduced class of functions called "Friedman-Ramanujan functions". We relate this claim to the study of the spectral gap of the Laplace-Beltrami operator, and prove it when TT fills a surface of Euler characteristic 00 or 1-1, providing a method to explicitly compute all coefficients in the second-order expansion. We conclude by displaying how the presence of tangles (which is an event of vanishing probability) prevents the sum over all types to satisfy the Friedman-Ramanujan property at the second order.

Keywords

Cite

@article{arxiv.2304.02678,
  title  = {Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps I},
  author = {Nalini Anantharaman and Laura Monk},
  journal= {arXiv preprint arXiv:2304.02678},
  year   = {2026}
}

Comments

71 pages, 18 figures. This new version is shorter due to making the former last section into a standalone article (to appear on arxiv). Content is otherwise unchanged

R2 v1 2026-06-28T09:51:38.944Z